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Chapter 3: Problem 62

Show that (a) \(E[X Y \mid Y=y]=y E[X \mid Y=y]\) (b) \(E[g(X, Y) \mid Y=y]=E[g(X, y) \mid Y=y]\) (c) \(E[X Y]=E[Y E[X \mid Y]]\)

Short Answer

Expert verified
We showed the following properties of conditional expectation: (a) \(E[X Y \mid Y = y] = y E[X \mid Y = y]\), by factoring out the constant term y from the summation. (b) \(E[g(X, Y) \mid Y = y] = E[g(X, y) \mid Y = y]\), as the summation is over x and y is treated as a constant. (c) \(E[X Y] = E[Y E[X \mid Y]]\), using the law of total expectation and the result from part (a).

Step by step solution

01

Write down the definition of conditional expectation

Using the definition of the conditional expectation: \(E[X Y \mid Y = y] = \sum\limits_{x \in X} xy \cdot P(X = x \mid Y = y)\)
02

Manipulate the summation

We can factor out the constant term y from the summation: \(E[X Y \mid Y = y] = y \cdot \sum\limits_{x \in X} x \cdot P(X = x \mid Y = y)\)
03

Recognize the result

The summation on the right side is the definition of the conditional expectation: \(y E[X \mid Y = y] = y \cdot \sum\limits_{x \in X} x \cdot P(X = x \mid Y = y)\) Hence, we showed that \(E[X Y \mid Y = y] = y E[X \mid Y = y]\) (b) Show that \(E[g(X, Y) \mid Y = y] = E[g(X, y) \mid Y = y]\)
04

Write down the definition of conditional expectation

Using the definition of the conditional expectation: \(E[g(X, Y) \mid Y = y] = \sum\limits_{x \in X} g(x, y) \cdot P(X = x \mid Y = y)\)
05

Recognize the result

Notice that the summation is over x, and inside the summation, y is treated as a constant: \(E[g(X, y) \mid Y = y] = \sum\limits_{x \in X} g(x, y) \cdot P(X = x \mid Y = y)\) Hence, we showed that \(E[g(X, Y) \mid Y = y] = E[g(X, y) \mid Y = y]\) (c) Show that \(E[X Y] = E[Y E[X \mid Y]]\)
06

Write down the definition of expectation over Y

Start with the definition of the expectation of a function of Y: \(E[Y E[X \mid Y]] = \sum\limits_{y \in Y} y E[X \mid Y = y] \cdot P(Y = y)\)
07

Use the result from part (a)

Replace \(y E[X \mid Y = y]\) with \(E[XY \mid Y=y]\) from part (a): \(E[Y E[X \mid Y]] = \sum\limits_{y \in Y} E[XY \mid Y = y] \cdot P(Y = y)\)
08

Apply the law of total expectation

Use the law of total expectation to compute the expectation of XY: \(E[X Y] = \sum\limits_{y \in Y} \sum\limits_{x \in X} xy \cdot P(X = x, Y = y)\) Notice that this sum can also be written as: \(E[X Y] = \sum\limits_{y \in Y}E[XY \mid Y = y] \cdot P(Y = y)\) Hence, we have shown that \(E[X Y] = E[Y E[X \mid Y]]\).

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